Since the discovery of quantum Hall effect in the 1980s, the topological band theory has been extensively developed and applied in various systems, ranging from insulators and semimetals to superconductors [1]}, [2]}, [3]}, [4]}. In the theory, the Chern number captures the winding of the eigenstates and is defined via the integral of the Berry curvature over the first Brillouin zone. It can not only be used to classify topological materials, but also quantify
the response of the system to an external field. For example, the Hall conductance is quantized and is proportional to the sum of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) invariants(or Chern numbers) of all filled bands [5]}, [6]}, [7]}. This theory was previously established in closed systems, and one may wonder if it holds valid for open systems.
